The river starts in the West Virginia panhandle, at the junction of the North Branch and the South Branch near Green Spring. The North Branch is the larger of the two, running further west into the West Virginia hills. The Jennings Randolph Lake, an important emergency water source for the Washington area, is on the North Branch. The South Branch follows the Allegheny south into West Virginia, winding through a sometimes very steep valley until it reaches its source in the Monongahela National Forest, near Spruce Knob.

I have seen many maps which delineate the boundary between
Maryland and Washington, DC on one side and Virginia and West Virginia
on the other as going down the center of the Potomac River. These
maps are incorrect.

"...all that tract of land...unto the true Meridian of the
first fountain of the River of Pattowmeck, and from thence trending
toward the South unto the farther banke of the fore-said river, and following
the West and South side thereof unto a certaine place called Cinquack,
situate neere the mouth of the said River, where it falls into the bay of Chesopeack..."

making the south bank of the river the boundary, putting the entire river
inside Maryland. This description was open to some interpretation:

The "first fountain" of the Potomac had not actually been located at the
time.

The Potomac divides into a "North Branch" and a "South Branch" a little
bit below Cumberland. A considerable amount of territory (4,300
square miles) lay between the two.

The lower Potomac is actually an estuary, and its water level rises and
falls (and its banks consequently shift) with the tide. Tributary
estuaries come in from Virginia and the points where one leaves off and
the main river begins are vague at best.

These and other factors caused boundary disputes between Maryland and Virginia
(and later West Virginia) which were not resolved until 1908. Maryland
had been founded as a haven for Roman Catholics, and early on, anti-Catholic
sentiment guaranteed that Maryland was on the losing end of these disputes.
Most importantly, in 1736 a stone was placed on the North
Branch of the Potomac and declared the river's "first fountain".

This boundary's definition has resulted in some odd situations in the
20th Century:

In the early 20th Century3, WESTVACO diverted the Potomac to the other side of its paper mill, suddenly putting the town of West Piedmont, West Virginia on the north side. It took a ruling of the Supreme Court to declare that West Piedmont, West Virginia was now Luke, Maryland.

During the first half of the 20th Century, Maryland was actually a haven
for gambling. Slot machines were illegal in Virginia, but enterprising
Virginians with waterfront property took advantage of the border's definition
by building long piers out into the river and building casinos on the piers.
The authorities could do nothing as the casinos were legally in Maryland.

Maryland and Virginia still dispute use of the lower Potomac (now, mostly
for fishing, oystering and crabbing) to this day.
1http://www.mdarchives.state.md.us/msa/educ/exhibits/founding/html/charter.html
2granted, Charles didn't own it to begin with, but that never stopped him.
3Luke, Maryland was chartered in 1922; I'm guessing it happened
then or the previous year.